A Unified Formula for the nth Derivative and the nth Anti-Derivative of the Power-Logarithmic Class

We give a complete solution to the problem of finding the nth derivative and the nth anti-derivative, where n is a real number or a symbol, of elementary and special classes of functions. In general, the solutions are given through unified formulas in terms of the Fox H-function which in many cases can be simplified for less general functions. In this work, we consider the class of the power-logarithmic class { f(x):f(x)=Sigma
_j=1

^lscr
p
_j
(x
<sup>alpha

j</sup>
)ln(beta
_j
x
<sup>gamma

j</sup>
+1)} (1) where alpha
_j
isinCopf, beta
_j
isinCopf\{0}, gamma
_j
isinRopf\{0}, and p
_j
's are polynomials of certain degrees.One of the key points in this work is that the approach does not depend on integration techniques. The arbitrary order of differentiation is found according to the Riemann-Liouville definition, whereas the generalized Cauchy n-fold integral is adopted for arbitrary order of integration. A software exhibition will be within the talk using the computer algebra system Maple.